/*
* INSANE - Interactive Structural Analysis Environment
*
* Copyright (C) 2003-2004
* Universidade Federal de Minas Gerais
* Escola de Engenharia
* Departamento de Engenharia de Estruturas
*
* Author's email : insane@dees.ufmg.br
* Author's Website : http://www.dees.ufmg.br/insane
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*
*/
package br.ufmg.dees.insane.util.xfem;

import br.ufmg.dees.insane.util.IMatrix;

/** Eigenvalues and eigenvectors of a real matrix. 
<P>
    If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
    diagonal and the eigenvector matrix V is orthogonal.
    I.e. A = V.times(D.times(V.transpose())) and 
    V.times(V.transpose()) equals the identity matrix.
<P>
    If A is not symmetric, then the eigenvalue matrix D is block diagonal
    with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
    lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The
    columns of V represent the eigenvectors in the sense that A*V = V*D,
    i.e. A.times(V) equals V.times(D).  The matrix V may be badly
    conditioned, or even singular, so the validity of the equation
    A = V*D*inverse(V) depends upon V.cond().
**/

/**
 * @author http://math.nist.gov/javanumerics/jama/
 *
 */
public class IEigenvalueDecomposition implements java.io.Serializable{
	
	private static final long serialVersionUID = 1L;

	/* ------------------------
	   Class variables
	 * ------------------------ */

	   /** Row and column dimension (square matrix).
	   @serial matrix dimension.
	   */
	   private int n;

	   /** Symmetry flag.
	   @serial internal symmetry flag.
	   */
	   private boolean issymmetric;

	   /** Arrays for internal storage of eigenvalues.
	   @serial internal storage of eigenvalues.
	   */
	   private double[] d, e;

	   /** Array for internal storage of eigenvectors.
	   @serial internal storage of eigenvectors.
	   */
	   private double[][] V;

	   /** Array for internal storage of nonsymmetric Hessenberg form.
	   @serial internal storage of nonsymmetric Hessenberg form.
	   */
	   private double[][] H;

	   /** Working storage for nonsymmetric algorithm.
	   @serial working storage for nonsymmetric algorithm.
	   */
	   private double[] ort;

	/* ------------------------
	   Private Methods
	 * ------------------------ */

	   // Symmetric Householder reduction to tridiagonal form.

	   private void tred2 () {

	   //  This is derived from the Algol procedures tred2 by
	   //  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
	   //  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
	   //  Fortran subroutine in EISPACK.

	      for (int j = 0; j < n; j++) {
	         d[j] = V[n-1][j];
	      }

	      // Householder reduction to tridiagonal form.
	   
	      for (int i = n-1; i > 0; i--) {
	   
	         // Scale to avoid under/overflow.
	   
	         double scale = 0.0;
	         double h = 0.0;
	         for (int k = 0; k < i; k++) {
	            scale = scale + Math.abs(d[k]);
	         }
	         if (scale == 0.0) {
	            e[i] = d[i-1];
	            for (int j = 0; j < i; j++) {
	               d[j] = V[i-1][j];
	               V[i][j] = 0.0;
	               V[j][i] = 0.0;
	            }
	         } else {
	   
	            // Generate Householder vector.
	   
	            for (int k = 0; k < i; k++) {
	               d[k] /= scale;
	               h += d[k] * d[k];
	            }
	            double f = d[i-1];
	            double g = Math.sqrt(h);
	            if (f > 0) {
	               g = -g;
	            }
	            e[i] = scale * g;
	            h = h - f * g;
	            d[i-1] = f - g;
	            for (int j = 0; j < i; j++) {
	               e[j] = 0.0;
	            }
	   
	            // Apply similarity transformation to remaining columns.
	   
	            for (int j = 0; j < i; j++) {
	               f = d[j];
	               V[j][i] = f;
	               g = e[j] + V[j][j] * f;
	               for (int k = j+1; k <= i-1; k++) {
	                  g += V[k][j] * d[k];
	                  e[k] += V[k][j] * f;
	               }
	               e[j] = g;
	            }
	            f = 0.0;
	            for (int j = 0; j < i; j++) {
	               e[j] /= h;
	               f += e[j] * d[j];
	            }
	            double hh = f / (h + h);
	            for (int j = 0; j < i; j++) {
	               e[j] -= hh * d[j];
	            }
	            for (int j = 0; j < i; j++) {
	               f = d[j];
	               g = e[j];
	               for (int k = j; k <= i-1; k++) {
	                  V[k][j] -= (f * e[k] + g * d[k]);
	               }
	               d[j] = V[i-1][j];
	               V[i][j] = 0.0;
	            }
	         }
	         d[i] = h;
	      }
	   
	      // Accumulate transformations.
	   
	      for (int i = 0; i < n-1; i++) {
	         V[n-1][i] = V[i][i];
	         V[i][i] = 1.0;
	         double h = d[i+1];
	         if (h != 0.0) {
	            for (int k = 0; k <= i; k++) {
	               d[k] = V[k][i+1] / h;
	            }
	            for (int j = 0; j <= i; j++) {
	               double g = 0.0;
	               for (int k = 0; k <= i; k++) {
	                  g += V[k][i+1] * V[k][j];
	               }
	               for (int k = 0; k <= i; k++) {
	                  V[k][j] -= g * d[k];
	               }
	            }
	         }
	         for (int k = 0; k <= i; k++) {
	            V[k][i+1] = 0.0;
	         }
	      }
	      for (int j = 0; j < n; j++) {
	         d[j] = V[n-1][j];
	         V[n-1][j] = 0.0;
	      }
	      V[n-1][n-1] = 1.0;
	      e[0] = 0.0;
	   } 

	   // Symmetric tridiagonal QL algorithm.
	   
	   private void tql2 () {

	   //  This is derived from the Algol procedures tql2, by
	   //  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
	   //  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
	   //  Fortran subroutine in EISPACK.
	   
	      for (int i = 1; i < n; i++) {
	         e[i-1] = e[i];
	      }
	      e[n-1] = 0.0;
	   
	      double f = 0.0;
	      double tst1 = 0.0;
	      double eps = Math.pow(2.0,-52.0);
	      for (int l = 0; l < n; l++) {

	         // Find small subdiagonal element
	   
	         tst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));
	         int m = l;
	         while (m < n) {
	            if (Math.abs(e[m]) <= eps*tst1) {
	               break;
	            }
	            m++;
	         }
	   
	         // If m == l, d[l] is an eigenvalue,
	         // otherwise, iterate.
	   
	         if (m > l) {
	            int iter = 0;
	            do {
	               iter = iter + 1;  // (Could check iteration count here.)
	   
	               // Compute implicit shift
	   
	               double g = d[l];
	               double p = (d[l+1] - g) / (2.0 * e[l]);
	               double r = IEigenvalueDecomposition.hypot(p,1.0);
	               if (p < 0) {
	                  r = -r;
	               }
	               d[l] = e[l] / (p + r);
	               d[l+1] = e[l] * (p + r);
	               double dl1 = d[l+1];
	               double h = g - d[l];
	               for (int i = l+2; i < n; i++) {
	                  d[i] -= h;
	               }
	               f = f + h;
	   
	               // Implicit QL transformation.
	   
	               p = d[m];
	               double c = 1.0;
	               double c2 = c;
	               double c3 = c;
	               double el1 = e[l+1];
	               double s = 0.0;
	               double s2 = 0.0;
	               for (int i = m-1; i >= l; i--) {
	                  c3 = c2;
	                  c2 = c;
	                  s2 = s;
	                  g = c * e[i];
	                  h = c * p;
	                  r = IEigenvalueDecomposition.hypot(p,e[i]);
	                  e[i+1] = s * r;
	                  s = e[i] / r;
	                  c = p / r;
	                  p = c * d[i] - s * g;
	                  d[i+1] = h + s * (c * g + s * d[i]);
	   
	                  // Accumulate transformation.
	   
	                  for (int k = 0; k < n; k++) {
	                     h = V[k][i+1];
	                     V[k][i+1] = s * V[k][i] + c * h;
	                     V[k][i] = c * V[k][i] - s * h;
	                  }
	               }
	               p = -s * s2 * c3 * el1 * e[l] / dl1;
	               e[l] = s * p;
	               d[l] = c * p;
	   
	               // Check for convergence.
	   
	            } while (Math.abs(e[l]) > eps*tst1);
	         }
	         d[l] = d[l] + f;
	         e[l] = 0.0;
	      }
	     
	      // Sort eigenvalues and corresponding vectors.
	   
	      for (int i = 0; i < n-1; i++) {
	         int k = i;
	         double p = d[i];
	         for (int j = i+1; j < n; j++) {
	            if (d[j] < p) {
	               k = j;
	               p = d[j];
	            }
	         }
	         if (k != i) {
	            d[k] = d[i];
	            d[i] = p;
	            for (int j = 0; j < n; j++) {
	               p = V[j][i];
	               V[j][i] = V[j][k];
	               V[j][k] = p;
	            }
	         }
	      }
	   }

	   // Nonsymmetric reduction to Hessenberg form.

	   private void orthes () {
	   
	      //  This is derived from the Algol procedures orthes and ortran,
	      //  by Martin and Wilkinson, Handbook for Auto. Comp.,
	      //  Vol.ii-Linear Algebra, and the corresponding
	      //  Fortran subroutines in EISPACK.
	   
	      int low = 0;
	      int high = n-1;
	   
	      for (int m = low+1; m <= high-1; m++) {
	   
	         // Scale column.
	   
	         double scale = 0.0;
	         for (int i = m; i <= high; i++) {
	            scale = scale + Math.abs(H[i][m-1]);
	         }
	         if (scale != 0.0) {
	   
	            // Compute Householder transformation.
	   
	            double h = 0.0;
	            for (int i = high; i >= m; i--) {
	               ort[i] = H[i][m-1]/scale;
	               h += ort[i] * ort[i];
	            }
	            double g = Math.sqrt(h);
	            if (ort[m] > 0) {
	               g = -g;
	            }
	            h = h - ort[m] * g;
	            ort[m] = ort[m] - g;
	   
	            // Apply Householder similarity transformation
	            // H = (I-u*u'/h)*H*(I-u*u')/h)
	   
	            for (int j = m; j < n; j++) {
	               double f = 0.0;
	               for (int i = high; i >= m; i--) {
	                  f += ort[i]*H[i][j];
	               }
	               f = f/h;
	               for (int i = m; i <= high; i++) {
	                  H[i][j] -= f*ort[i];
	               }
	           }
	   
	           for (int i = 0; i <= high; i++) {
	               double f = 0.0;
	               for (int j = high; j >= m; j--) {
	                  f += ort[j]*H[i][j];
	               }
	               f = f/h;
	               for (int j = m; j <= high; j++) {
	                  H[i][j] -= f*ort[j];
	               }
	            }
	            ort[m] = scale*ort[m];
	            H[m][m-1] = scale*g;
	         }
	      }
	   
	      // Accumulate transformations (Algol's ortran).

	      for (int i = 0; i < n; i++) {
	         for (int j = 0; j < n; j++) {
	            V[i][j] = (i == j ? 1.0 : 0.0);
	         }
	      }

	      for (int m = high-1; m >= low+1; m--) {
	         if (H[m][m-1] != 0.0) {
	            for (int i = m+1; i <= high; i++) {
	               ort[i] = H[i][m-1];
	            }
	            for (int j = m; j <= high; j++) {
	               double g = 0.0;
	               for (int i = m; i <= high; i++) {
	                  g += ort[i] * V[i][j];
	               }
	               // Double division avoids possible underflow
	               g = (g / ort[m]) / H[m][m-1];
	               for (int i = m; i <= high; i++) {
	                  V[i][j] += g * ort[i];
	               }
	            }
	         }
	      }
	   }


	   // Complex scalar division.

	   private transient double cdivr, cdivi;
	   private void cdiv(double xr, double xi, double yr, double yi) {
	      double r,d;
	      if (Math.abs(yr) > Math.abs(yi)) {
	         r = yi/yr;
	         d = yr + r*yi;
	         cdivr = (xr + r*xi)/d;
	         cdivi = (xi - r*xr)/d;
	      } else {
	         r = yr/yi;
	         d = yi + r*yr;
	         cdivr = (r*xr + xi)/d;
	         cdivi = (r*xi - xr)/d;
	      }
	   }


	   // Nonsymmetric reduction from Hessenberg to real Schur form.

	   private void hqr2 () {
	   
	      //  This is derived from the Algol procedure hqr2,
	      //  by Martin and Wilkinson, Handbook for Auto. Comp.,
	      //  Vol.ii-Linear Algebra, and the corresponding
	      //  Fortran subroutine in EISPACK.
	   
	      // Initialize
	   
	      int nn = this.n;
	      int n = nn-1;
	      int low = 0;
	      int high = nn-1;
	      double eps = Math.pow(2.0,-52.0);
	      double exshift = 0.0;
	      double p=0,q=0,r=0,s=0,z=0,t,w,x,y;
	   
	      // Store roots isolated by balanc and compute matrix norm
	   
	      double norm = 0.0;
	      for (int i = 0; i < nn; i++) {
	         if (i < low | i > high) {
	            d[i] = H[i][i];
	            e[i] = 0.0;
	         }
	         for (int j = Math.max(i-1,0); j < nn; j++) {
	            norm = norm + Math.abs(H[i][j]);
	         }
	      }
	   
	      // Outer loop over eigenvalue index
	   
	      int iter = 0;
	      while (n >= low) {
	   
	         // Look for single small sub-diagonal element
	   
	         int l = n;
	         while (l > low) {
	            s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);
	            if (s == 0.0) {
	               s = norm;
	            }
	            if (Math.abs(H[l][l-1]) < eps * s) {
	               break;
	            }
	            l--;
	         }
	       
	         // Check for convergence
	         // One root found
	   
	         if (l == n) {
	            H[n][n] = H[n][n] + exshift;
	            d[n] = H[n][n];
	            e[n] = 0.0;
	            n--;
	            iter = 0;
	   
	         // Two roots found
	   
	         } else if (l == n-1) {
	            w = H[n][n-1] * H[n-1][n];
	            p = (H[n-1][n-1] - H[n][n]) / 2.0;
	            q = p * p + w;
	            z = Math.sqrt(Math.abs(q));
	            H[n][n] = H[n][n] + exshift;
	            H[n-1][n-1] = H[n-1][n-1] + exshift;
	            x = H[n][n];
	   
	            // Real pair
	   
	            if (q >= 0) {
	               if (p >= 0) {
	                  z = p + z;
	               } else {
	                  z = p - z;
	               }
	               d[n-1] = x + z;
	               d[n] = d[n-1];
	               if (z != 0.0) {
	                  d[n] = x - w / z;
	               }
	               e[n-1] = 0.0;
	               e[n] = 0.0;
	               x = H[n][n-1];
	               s = Math.abs(x) + Math.abs(z);
	               p = x / s;
	               q = z / s;
	               r = Math.sqrt(p * p+q * q);
	               p = p / r;
	               q = q / r;
	   
	               // Row modification
	   
	               for (int j = n-1; j < nn; j++) {
	                  z = H[n-1][j];
	                  H[n-1][j] = q * z + p * H[n][j];
	                  H[n][j] = q * H[n][j] - p * z;
	               }
	   
	               // Column modification
	   
	               for (int i = 0; i <= n; i++) {
	                  z = H[i][n-1];
	                  H[i][n-1] = q * z + p * H[i][n];
	                  H[i][n] = q * H[i][n] - p * z;
	               }
	   
	               // Accumulate transformations
	   
	               for (int i = low; i <= high; i++) {
	                  z = V[i][n-1];
	                  V[i][n-1] = q * z + p * V[i][n];
	                  V[i][n] = q * V[i][n] - p * z;
	               }
	   
	            // Complex pair
	   
	            } else {
	               d[n-1] = x + p;
	               d[n] = x + p;
	               e[n-1] = z;
	               e[n] = -z;
	            }
	            n = n - 2;
	            iter = 0;
	   
	         // No convergence yet
	   
	         } else {
	   
	            // Form shift
	   
	            x = H[n][n];
	            y = 0.0;
	            w = 0.0;
	            if (l < n) {
	               y = H[n-1][n-1];
	               w = H[n][n-1] * H[n-1][n];
	            }
	   
	            // Wilkinson's original ad hoc shift
	   
	            if (iter == 10) {
	               exshift += x;
	               for (int i = low; i <= n; i++) {
	                  H[i][i] -= x;
	               }
	               s = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-2]);
	               x = y = 0.75 * s;
	               w = -0.4375 * s * s;
	            }

	            // MATLAB's new ad hoc shift

	            if (iter == 30) {
	                s = (y - x) / 2.0;
	                s = s * s + w;
	                if (s > 0) {
	                    s = Math.sqrt(s);
	                    if (y < x) {
	                       s = -s;
	                    }
	                    s = x - w / ((y - x) / 2.0 + s);
	                    for (int i = low; i <= n; i++) {
	                       H[i][i] -= s;
	                    }
	                    exshift += s;
	                    x = y = w = 0.964;
	                }
	            }
	   
	            iter = iter + 1;   // (Could check iteration count here.)
	   
	            // Look for two consecutive small sub-diagonal elements
	   
	            int m = n-2;
	            while (m >= l) {
	               z = H[m][m];
	               r = x - z;
	               s = y - z;
	               p = (r * s - w) / H[m+1][m] + H[m][m+1];
	               q = H[m+1][m+1] - z - r - s;
	               r = H[m+2][m+1];
	               s = Math.abs(p) + Math.abs(q) + Math.abs(r);
	               p = p / s;
	               q = q / s;
	               r = r / s;
	               if (m == l) {
	                  break;
	               }
	               if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
	                  eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
	                  Math.abs(H[m+1][m+1])))) {
	                     break;
	               }
	               m--;
	            }
	   
	            for (int i = m+2; i <= n; i++) {
	               H[i][i-2] = 0.0;
	               if (i > m+2) {
	                  H[i][i-3] = 0.0;
	               }
	            }
	   
	            // Double QR step involving rows l:n and columns m:n
	   
	            for (int k = m; k <= n-1; k++) {
	               boolean notlast = (k != n-1);
	               if (k != m) {
	                  p = H[k][k-1];
	                  q = H[k+1][k-1];
	                  r = (notlast ? H[k+2][k-1] : 0.0);
	                  x = Math.abs(p) + Math.abs(q) + Math.abs(r);
	                  if (x != 0.0) {
	                     p = p / x;
	                     q = q / x;
	                     r = r / x;
	                  }
	               }
	               if (x == 0.0) {
	                  break;
	               }
	               s = Math.sqrt(p * p + q * q + r * r);
	               if (p < 0) {
	                  s = -s;
	               }
	               if (s != 0) {
	                  if (k != m) {
	                     H[k][k-1] = -s * x;
	                  } else if (l != m) {
	                     H[k][k-1] = -H[k][k-1];
	                  }
	                  p = p + s;
	                  x = p / s;
	                  y = q / s;
	                  z = r / s;
	                  q = q / p;
	                  r = r / p;
	   
	                  // Row modification
	   
	                  for (int j = k; j < nn; j++) {
	                     p = H[k][j] + q * H[k+1][j];
	                     if (notlast) {
	                        p = p + r * H[k+2][j];
	                        H[k+2][j] = H[k+2][j] - p * z;
	                     }
	                     H[k][j] = H[k][j] - p * x;
	                     H[k+1][j] = H[k+1][j] - p * y;
	                  }
	   
	                  // Column modification
	   
	                  for (int i = 0; i <= Math.min(n,k+3); i++) {
	                     p = x * H[i][k] + y * H[i][k+1];
	                     if (notlast) {
	                        p = p + z * H[i][k+2];
	                        H[i][k+2] = H[i][k+2] - p * r;
	                     }
	                     H[i][k] = H[i][k] - p;
	                     H[i][k+1] = H[i][k+1] - p * q;
	                  }
	   
	                  // Accumulate transformations
	   
	                  for (int i = low; i <= high; i++) {
	                     p = x * V[i][k] + y * V[i][k+1];
	                     if (notlast) {
	                        p = p + z * V[i][k+2];
	                        V[i][k+2] = V[i][k+2] - p * r;
	                     }
	                     V[i][k] = V[i][k] - p;
	                     V[i][k+1] = V[i][k+1] - p * q;
	                  }
	               }  // (s != 0)
	            }  // k loop
	         }  // check convergence
	      }  // while (n >= low)
	      
	      // Backsubstitute to find vectors of upper triangular form

	      if (norm == 0.0) {
	         return;
	      }
	   
	      for (n = nn-1; n >= 0; n--) {
	         p = d[n];
	         q = e[n];
	   
	         // Real vector
	   
	         if (q == 0) {
	            int l = n;
	            H[n][n] = 1.0;
	            for (int i = n-1; i >= 0; i--) {
	               w = H[i][i] - p;
	               r = 0.0;
	               for (int j = l; j <= n; j++) {
	                  r = r + H[i][j] * H[j][n];
	               }
	               if (e[i] < 0.0) {
	                  z = w;
	                  s = r;
	               } else {
	                  l = i;
	                  if (e[i] == 0.0) {
	                     if (w != 0.0) {
	                        H[i][n] = -r / w;
	                     } else {
	                        H[i][n] = -r / (eps * norm);
	                     }
	   
	                  // Solve real equations
	   
	                  } else {
	                     x = H[i][i+1];
	                     y = H[i+1][i];
	                     q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
	                     t = (x * s - z * r) / q;
	                     H[i][n] = t;
	                     if (Math.abs(x) > Math.abs(z)) {
	                        H[i+1][n] = (-r - w * t) / x;
	                     } else {
	                        H[i+1][n] = (-s - y * t) / z;
	                     }
	                  }
	   
	                  // Overflow control
	   
	                  t = Math.abs(H[i][n]);
	                  if ((eps * t) * t > 1) {
	                     for (int j = i; j <= n; j++) {
	                        H[j][n] = H[j][n] / t;
	                     }
	                  }
	               }
	            }
	   
	         // Complex vector
	   
	         } else if (q < 0) {
	            int l = n-1;

	            // Last vector component imaginary so matrix is triangular
	   
	            if (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {
	               H[n-1][n-1] = q / H[n][n-1];
	               H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
	            } else {
	               cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
	               H[n-1][n-1] = cdivr;
	               H[n-1][n] = cdivi;
	            }
	            H[n][n-1] = 0.0;
	            H[n][n] = 1.0;
	            for (int i = n-2; i >= 0; i--) {
	               double ra,sa,vr,vi;
	               ra = 0.0;
	               sa = 0.0;
	               for (int j = l; j <= n; j++) {
	                  ra = ra + H[i][j] * H[j][n-1];
	                  sa = sa + H[i][j] * H[j][n];
	               }
	               w = H[i][i] - p;
	   
	               if (e[i] < 0.0) {
	                  z = w;
	                  r = ra;
	                  s = sa;
	               } else {
	                  l = i;
	                  if (e[i] == 0) {
	                     cdiv(-ra,-sa,w,q);
	                     H[i][n-1] = cdivr;
	                     H[i][n] = cdivi;
	                  } else {
	   
	                     // Solve complex equations
	   
	                     x = H[i][i+1];
	                     y = H[i+1][i];
	                     vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
	                     vi = (d[i] - p) * 2.0 * q;
	                     if (vr == 0.0 & vi == 0.0) {
	                        vr = eps * norm * (Math.abs(w) + Math.abs(q) +
	                        Math.abs(x) + Math.abs(y) + Math.abs(z));
	                     }
	                     cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
	                     H[i][n-1] = cdivr;
	                     H[i][n] = cdivi;
	                     if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
	                        H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
	                        H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
	                     } else {
	                        cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
	                        H[i+1][n-1] = cdivr;
	                        H[i+1][n] = cdivi;
	                     }
	                  }
	   
	                  // Overflow control

	                  t = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));
	                  if ((eps * t) * t > 1) {
	                     for (int j = i; j <= n; j++) {
	                        H[j][n-1] = H[j][n-1] / t;
	                        H[j][n] = H[j][n] / t;
	                     }
	                  }
	               }
	            }
	         }
	      }
	   
	      // Vectors of isolated roots
	   
	      for (int i = 0; i < nn; i++) {
	         if (i < low | i > high) {
	            for (int j = i; j < nn; j++) {
	               V[i][j] = H[i][j];
	            }
	         }
	      }
	   
	      // Back transformation to get eigenvectors of original matrix
	   
	      for (int j = nn-1; j >= low; j--) {
	         for (int i = low; i <= high; i++) {
	            z = 0.0;
	            for (int k = low; k <= Math.min(j,high); k++) {
	               z = z + V[i][k] * H[k][j];
	            }
	            V[i][j] = z;
	         }
	      }
	   }


	/* ------------------------
	   Constructor
	 * ------------------------ */

	   /** Check for symmetry, then construct the eigenvalue decomposition
	   @param A    Square matrix
	   @return     Structure to access D and V.
	   */

	   public IEigenvalueDecomposition (IMatrix mat) {
	      
	      n = mat.getNumRow();
	      V = new double[n][n];
	      d = new double[n];
	      e = new double[n];
	      double[][] A = new double[n][n];
	      for (int i = 0; i<n; i++)
	    	  for (int j = 0; j<n; j++)
	    		  A[i][j] = mat.getElement(i,j);

	      issymmetric = true;
	      for (int j = 0; (j < n) & issymmetric; j++) {
	         for (int i = 0; (i < n) & issymmetric; i++) {
	            issymmetric = (A[i][j] == A[j][i]);
	         }
	      }

	      if (issymmetric) {
	         for (int i = 0; i < n; i++) {
	            for (int j = 0; j < n; j++) {
	               V[i][j] = A[i][j];
	            }
	         }
	   
	         // Tridiagonalize.
	         tred2();
	   
	         // Diagonalize.
	         tql2();

	      } else {
	         H = new double[n][n];
	         ort = new double[n];
	         
	         for (int j = 0; j < n; j++) {
	            for (int i = 0; i < n; i++) {
	               H[i][j] = A[i][j];
	            }
	         }
	   
	         // Reduce to Hessenberg form.
	         orthes();
	   
	         // Reduce Hessenberg to real Schur form.
	         hqr2();
	      }
	   }

	/* ------------------------
	   Public Methods
	 * ------------------------ */

	   /** Return the eigenvector matrix
	   @return     V
	   */

	   public IMatrix getV () {
	      return new IMatrix(V);
	   }

	   /** Return the real parts of the eigenvalues
	   @return     real(diag(D))
	   */

	   public double[] getRealEigenvalues () {
	      return d;
	   }
	   
	   /** Return the imaginary parts of the eigenvalues
	   @return     imag(diag(D))
	   */

	   public double[] getImagEigenvalues () {
	      return e;
	   }

	   /** Return the block diagonal eigenvalue matrix
	   @return     D
	   */

	   public IMatrix getD () {
	      double[][] D = new double[n][n];
	      for (int i = 0; i < n; i++) {
	         for (int j = 0; j < n; j++) {
	            D[i][j] = 0.0;
	         }
	         D[i][i] = d[i];
	         if (e[i] > 0) {
	            D[i][i+1] = e[i];
	         } else if (e[i] < 0) {
	            D[i][i-1] = e[i];
	         }
	      }
	      IMatrix X = new IMatrix(D);
	      return X;
	   }

	/* ------------------------
	   Static Methods
	 * ------------------------ */
	   
	   /** sqrt(a^2 + b^2) without under/overflow. **/

	   public static double hypot(double a, double b) {

	      double r;

	      if (Math.abs(a) > Math.abs(b)) {

	         r = b/a;

	         r = Math.abs(a)*Math.sqrt(1+r*r);

	      } else if (b != 0) {

	         r = a/b;

	         r = Math.abs(b)*Math.sqrt(1+r*r);

	      } else {

	         r = 0.0;

	      }

	      return r;

	   }

}
